Reduced superschemes and the combinatorics of toric supervarieties

We propose new definitions of integral, reduced, and normal superrings and superschemes to properly establish the notion of a supervariety. We generalize several results about classical reduced rings and varieties to the supergeometric setting, including an equivalence of categories between certain toric supervarieties and decorated polyhedral fans. These decorated fans are shown to encode important geometric information about the corresponding toric supervarieties. We then investigate some naturally-occurring toric supervarieties inside the isomeric supergrassmannian, which we show admits a nice description as a decorated polytope.